Question 174362
Hi, Hope I can help,
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Find an equation of the line that is parallel to the line {{{ y= (-4/3)x+7 }}} and that passes through the point (7,3)
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We are trying to find a line parallel to this line
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{{{ graph ( 800, 800, -20, 20, -20, 20, (-4/3)x+7 ) }}}
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lines are given in the form {{{ y = mx + b }}}, where "m" is the slope, and "b" is the y intercept
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{{{ y= (-4/3)x+7 }}}, this line has a slope of {{{ -4/3 }}}, and y intercept of "7"
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We are trying to find a parallel line, so that means the line will have the same slope as the other, or {{{ -4/3 }}}
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If we took the line form {{{ y = mx + b }}}, we can start replacing the letters for our unknown line
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We know that "m" is {{{ -4/3 }}}, since it is parallel to the given line
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Our new formula is 
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{{{ y = (-4/3)x + b }}} ( we replaced "m" with the known slope {{{ -4/3 }}} )
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{{{ y = (-4/3)x + b }}}, we now have 3 variables, we will have to get rid of 2 of them
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We are also given a point, (7,3)
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Points are in the form of (x,y)
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(7,3):(x,y)
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 x = 7
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 y = 3
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{{{ y = (-4/3)x + b }}}, now replace the "x" and "y" with the numbers
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{{{ y = (-4/3)x + b }}} = {{{ (3) = (-4/3)(7) + b }}} = {{{ 3 = (-28/3) + b }}}
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Now we just need to solve for "b"
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{{{ 3 = (-28/3) + b }}}, we will move the {{{ -28/3 }}} over to the left side
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{{{ 3 = (-28/3) + b }}} = {{{ 3 + (28/3) = (-28/3) + (28/3) + b }}} = {{{ 3 + (28/3) = b }}} = {{{ 3 + (28/3) = b }}} = {{{ 37/3 = b }}}, or {{{ b = 37/3 }}}
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You can check your answer by replacing "b" with {{{ 37/3 }}} in the equation
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{{{ 3 = (-28/3) + b }}} = {{{ 3 = (-28/3) + (37/3) }}} = {{{ 3 = 9/3 }}} = {{{ 3 = 3 }}} (True)
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Now just replace "b" in our original equation ( one where we had "x" and "y" )
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{{{ y = (-4/3)x + b }}} = {{{ y = (-4/3)x + (37/3) }}}
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{{{ y = (-4/3)x + (37/3) }}}  is the line in point slope form, if you want it in standard form {{{ ax + by = c }}}, here is how you would convert it
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{{{ y = (-4/3)x + (37/3) }}}, you would first multiply each side by "3"
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{{{ y = (-4/3)x + (37/3) }}} = {{{ 3(y) = 3((-4/3)x + (37/3)) }}}, using distribution
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{{{ 3y = highlight(3)(highlight((-4/3)x) + (37/3)) }}} = {{{ 3y = highlight(3)((-4/3)x + highlight((37/3))) }}} = {{{ 3y = 3((-4/3)x) + 3(37/3) }}}
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{{{ 3y = 3((-4/3)x) + 3(37/3) }}} = {{{ 3y = -4x + 37 }}}
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Now we need to move (-4x) to the left side
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{{{ 3y = -4x + 37 }}} = {{{ 3y +4x = -4x + 4x + 37 }}} = {{{ 3y + 4x = 37 }}}
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Rearranging the left side
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{{{ 3y + 4x = 37 }}} = {{{4x + 3y = 37 }}}, this is the standard form of the equation
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If you want to see if a point is on a line you would replace "x" and "y" with the point (x,y)
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Our point is (7,3)(x,y)
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 x = 7
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 y = 3
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{{{4x + 3y = 37 }}} = {{{ 4(7) + 3(3) = 37 }}} =  {{{ 28 + 9 = 37 }}} = {{{ 37 = 37 }}} (True)
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The graph of this line is 
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{{{ graph ( 800, 800, -20, 20, -20, 20, (-4/3)x+(37/3) ) }}}
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It is parallel to the original line and has (7,3) as a point
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{{{ drawing ( 800,800,-20,20,-20,20, grid (1), circle (7,3,0.1), blue (circle (7,3,0.2)), graph ( 800, 800, -20, 20, -20, 20, (-4/3)x+(37/3), (-4/3)x+7 )) }}}
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Your line that you were looking for or your answer is
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Standard form: {{{4x + 3y = 37 }}}
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Point-Slope form: {{{ y = (-4/3)x + (37/3) }}}
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Hope I helped, Levi